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Counting Hamiltonian Cycles in 2-Tiled Graphs

Author

Listed:
  • Alen Vegi Kalamar

    (Department of Mathematics and Computer Science, University of Maribor, 2000 Maribor, Slovenia
    Comtrade Gaming, 2000 Maribor, Slovenia
    Crossing numbers and their applications (ARRS Project Number J1-8130).
    Structural, optimization, and algorithmic problems in geometric and topological graph representations (ARRS Project Number J1-2452).)

  • Tadej Žerak

    (Department of Mathematics and Computer Science, University of Maribor, 2000 Maribor, Slovenia
    DataBitLab, d.o.o., 2000 Maribor, Slovenia)

  • Drago Bokal

    (Department of Mathematics and Computer Science, University of Maribor, 2000 Maribor, Slovenia
    DataBitLab, d.o.o., 2000 Maribor, Slovenia
    Institute of Mathematics, Physics and Mechanics, University of Ljubljana, 1000 Ljubljana, Slovenia
    Crossing numbers and their applications (ARRS Project Number J1-8130).)

Abstract

In 1930, Kuratowski showed that K 3 , 3 and K 5 are the only two minor-minimal nonplanar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň and Kochol showed that there are infinitely many k -crossing-critical graphs for any k ≥ 2 , even if restricted to simple 3-connected graphs. Recently, 2-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodroža-Pantić, Kwong, Doroslovački, and Pantić.

Suggested Citation

  • Alen Vegi Kalamar & Tadej Žerak & Drago Bokal, 2021. "Counting Hamiltonian Cycles in 2-Tiled Graphs," Mathematics, MDPI, vol. 9(6), pages 1-27, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:6:p:693-:d:522672
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